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Find the integral $\int\frac{4x-5}{\left(x-1\right)\left(x^2+3\right)^2}dx$

Step-by-step Solution

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Final Answer

$-\frac{1}{16}\ln\left(x-1\right)+0.445041\arctan\left(0.577348x\right)+\frac{17x}{24\left(x^2+3\right)}+\frac{-1}{8\left(x^2+3\right)}-\frac{1}{16}\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$
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Step-by-step Solution

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Rewrite the fraction $\frac{4x-5}{\left(x-1\right)\left(x^2+3\right)^2}$ in $3$ simpler fractions using partial fraction decomposition

$\frac{4x-5}{\left(x-1\right)\left(x^2+3\right)^2}=\frac{A}{x-1}+\frac{Bx+C}{\left(x^2+3\right)^2}+\frac{Dx+F}{x^2+3}$

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$\frac{4x-5}{\left(x-1\right)\left(x^2+3\right)^2}=\frac{A}{x-1}+\frac{Bx+C}{\left(x^2+3\right)^2}+\frac{Dx+F}{x^2+3}$

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Learn how to solve problems step by step online. Find the integral int((4x-5)/((x-1)(x^2+3)^2))dx. Rewrite the fraction \frac{4x-5}{\left(x-1\right)\left(x^2+3\right)^2} in 3 simpler fractions using partial fraction decomposition. Find the values for the unknown coefficients: A, B, C, D, F. The first step is to multiply both sides of the equation from the previous step by \left(x-1\right)\left(x^2+3\right)^2. Multiply both sides of the equality by 1 to simplify the fractions. Multiplying polynomials.

Final Answer

$-\frac{1}{16}\ln\left(x-1\right)+0.445041\arctan\left(0.577348x\right)+\frac{17x}{24\left(x^2+3\right)}+\frac{-1}{8\left(x^2+3\right)}-\frac{1}{16}\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$

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Function Plot

Plotting: $-\frac{1}{16}\ln\left(x-1\right)+0.445041\arctan\left(0.577348x\right)+\frac{17x}{24\left(x^2+3\right)}+\frac{-1}{8\left(x^2+3\right)}-\frac{1}{16}\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$

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7
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9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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